Description of drip-line nuclei with GSM and Gamow/HFB frameworks

Nicolas MichelNicolas MichelImportance of continuum for nuclei close to Importance of continuum for nuclei close to drip-linedrip-line

May 20th, 2009May 20th, 2009

Description of drip-line Description of drip-line nuclei with GSM and nuclei with GSM and

Gamow/HFB frameworks Gamow/HFB frameworks

Nicolas MichelNicolas Michel (CEA/IRFU/SPhN)(CEA/IRFU/SPhN)Marek PloszajszakMarek Ploszajszak (GANIL) (GANIL)

Witek NazarewiczWitek Nazarewicz (UT/ORNL) (UT/ORNL)Kenichi MatsuyanagiKenichi Matsuyanagi (Kyoto University) (Kyoto University)

Mario StoitsovMario Stoitsov (ORNL – University of Tennessee) (ORNL – University of Tennessee)

May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines

Nicolas MichelNicolas Michel 22

Plan Plan

• Scientific motivation: drip-line nucleiScientific motivation: drip-line nuclei• Gamow states definition and normalization, Berggren basisGamow states definition and normalization, Berggren basis• Spectroscopic factors, overlap functionsSpectroscopic factors, overlap functions• 66He/He/55He, He, 77He/He/66He, He, 1818O*/O*/1717O*, O*, 66Be/Be/55LiLi

• Gamow HFB framework: HF basis diagonalization, direct integrationGamow HFB framework: HF basis diagonalization, direct integration• Applications: Nickel chainApplications: Nickel chain• PPööschl-Teller-Ginocchio (PTG) basis for loosely bound systemsschl-Teller-Ginocchio (PTG) basis for loosely bound systems• Applications: Nickel chain (spherical)Applications: Nickel chain (spherical) Zirconium and Magnesium (deformed)Zirconium and Magnesium (deformed)• Conclusion and perspectivesConclusion and perspectives



Scientific motivation Scientific motivation



Gamow statesGamow states

• Georg GamowGeorg Gamow : simple model of : simple model of decay decay G.A. Gamow, Zs f. Phys. G.A. Gamow, Zs f. Phys. 5151 (1928) 204; (1928) 204; 5252 (1928) 510 (1928) 510

• DefinitionDefinition : :

• Straightforward generalization to non-local potentials (HF)Straightforward generalization to non-local potentials (HF)


Nicolas MichelNicolas Michel

• NormalizationNormalization: : complex scalingcomplex scaling

• Analytic continuationAnalytic continuation: integral : integral independentindependent of R and of R and θθ

• Normalization of bound and resonant states: Normalization of bound and resonant states:

• Scattering states:Scattering states: normalization normalization impossibleimpossible with complex scaling with complex scaling Normalization with Dirac delta: Normalization with Dirac delta:

55

Complex scalingComplex scaling



Complex energy statesComplex energy states

boundbound

broad resonancesbroad resonances

narrow resonancesnarrow resonances

LL++ : arbitrary contour : arbitrary contour

antibound statesantibound states

capturing statescapturing states

Im(k)Im(k)

Re(k)Re(k)

Berggren completeness relationBerggren completeness relation



Completeness relations with Completeness relations with Gamow states Gamow states

• Berggren completeness relationBerggren completeness relation (l,j) : (l,j) :

T. Berggren, Nucl. Phys. A T. Berggren, Nucl. Phys. A 109109, (1967) 205, (1967) 205

• Continuum discretizationContinuum discretization : :



• O isotopesO isotopes : valence particles above : valence particles above 1616OO core : core : H = T + WS (H = T + WS (1717O) + SDIO) + SDI

0d0d5/25/2, 1s, 1s1/21/2 (bound), 0d (bound), 0d3/23/2 (resonant), d(resonant), d3/23/2 scattering continuum scattering continuum

• He and Be isotopesHe and Be isotopes : valence particles above : valence particles above 44HeHe core core

H = T + WS (H = T + WS (55He/He/55Li) + SGILi) + SGI

0p0p3/23/2, 0p, 0p1/21/2 (resonant), p (resonant), p3/23/2 and p and p1/21/2 scattering continuums scattering continuums

SGI : Surface Gaussian Interaction :SGI : Surface Gaussian Interaction :

88

Application : O, He and Be isotopesApplication : O, He and Be isotopes



Spectroscopic factors in GSMSpectroscopic factors in GSM

• One particle emission channelOne particle emission channel: (l,j,: (l,j,))

• Basis-independent definitionBasis-independent definition::

• ExperimentalExperimental : : cuspscusps appear in appear in cross-sectionscross-sections

• StandardStandard : representation : representation dependencedependence ( (nn,l,j,,l,j,))

• 55He / He / 66He, He, 55Li / Li / 66BeBe : non resonant components : non resonant components necessarynecessary..



Spectroscopic factorSpectroscopic factor::

Nucleon-nucleon Nucleon-nucleon correlationscorrelations measure measure

Universal behaviorUniversal behaviorclose to emission thresholdclose to emission threshold

N. Michel et al.,N. Michel et al.,Phys. Rev. C, (Rap. Comm.), Phys. Rev. C, (Rap. Comm.), 7575 031301(R) (2007) 031301(R) (2007)

VVSGISGI

variesvaries

VVSGISGI

variesvaries

WSWSvariesvaries



Real WS fitReal WS fit

Complex WS fitComplex WS fit

O(r)O(r)

N.Michel, W.Nazarewicz, M.Ploszajczak,N.Michel, W.Nazarewicz, M.Ploszajczak,

Nucl. Phys. A Nucl. Phys. A 794794 29 (2007) 29 (2007)



1212



WSWSvariesvaries



1313



VVSGISGI variesvaries



VVSGISGI variesvaries VVSGISGI variesvaries

Mirror reactions : Mirror reactions : 66He/He/55HeHe and and 66Be/Be/55LiLi

66He/He/55HeHe

66Be/Be/55LiLi





66Be/Be/55LiLi

66He/He/55HeHe



HFB frameworkHFB framework• HFB ground state HFB ground state : product of independent quasi-particles: product of independent quasi-particles

• HFB equations: HFB equations:

• Standard methods of resolutionStandard methods of resolution

HO basis diagonalizationHO basis diagonalization: : well bound states onlywell bound states only

THO basis diagonalizationTHO basis diagonalization: : basis dependence from scaling function ?basis dependence from scaling function ?

Direct integrationDirect integration: : very precisevery precise, , but long but long (box boundary conditions)(box boundary conditions)

1717



PTG/HFB and Gamow/HFB modelsPTG/HFB and Gamow/HFB models

• Continuous basis methodsContinuous basis methods

Complex-energy formalismComplex-energy formalism

Bound, resonant Bound, resonant andand scattering scattering states (Berggren basis)states (Berggren basis)

Berggren basis of Berggren basis of HF particle states HF particle states ((two-basis methodtwo-basis method) )

BerggrenBerggren quasi-particles states quasi-particles states calculated in coordinate spacecalculated in coordinate space

Real-energy formalismReal-energy formalism

Basis generated by a Basis generated by a Pöschl-Teller-Ginocchio (PTG) potentialPöschl-Teller-Ginocchio (PTG) potential

• Examples:Examples: Ni chain, Ni chain, 110110Zr and Zr and 4040MgMg, HFB density functional: , HFB density functional: Sly4 + surface pairing Sly4 + surface pairing

orbital momentum: l=0 to l=10, Eorbital momentum: l=0 to l=10, Ecutcut = 60 MeV = 60 MeV



Gamow HFB framework: Gamow HFB framework: HF basis diagonalizationHF basis diagonalization

• Two-basis methodTwo-basis method

BasisBasis generated bygenerated by ph part ph part of HFB hamiltonian: of HFB hamiltonian:

B. Gall et al., Z. Phys. B. Gall et al., Z. Phys. A348A348 183 (1994) 183 (1994)

• HFB matrix structure:HFB matrix structure:

• Diagonalization of HFB matrix in PTG or Gamow HF basisDiagonalization of HFB matrix in PTG or Gamow HF basis



Gamow HFB framework:Gamow HFB framework:direct integrationdirect integration

N. Michel, K.Matsuyanagi, M. StoitsovN. Michel, K.Matsuyanagi, M. StoitsovPhys. Rev. C, Phys. Rev. C, 7878 044319 (2008) 044319 (2008)



HF and PTG potentialsHF and PTG potentials

M. Stoitsov, N. Michel, K.Matsuyanagi,M. Stoitsov, N. Michel, K.Matsuyanagi,Phys. Rev. C, Phys. Rev. C, 7777, 054301 (2008), 054301 (2008)



HF/PTG wave functionsHF/PTG wave functions

r (fm)r (fm)

---- : PTG---- : PTG : HF: HF




densitiesdensities----- : prot.----- : prot. : neut.: neut.

… … : THO: THO


PTG PTG basisbasis

Long axisLong axis

Short axisShort axis




PairingPairingdensitiesdensities

----- : prot.----- : prot. : neut.: neut.

… … : THO: THO

PTG PTG basisbasis



Black solid line : Black solid line : HFB boxHFB box

Dashed green line :Dashed green line : GHFB coord. GHFB coord.

Dotted blue line :Dotted blue line : GHF basisGHF basis

Gamow HFBGamow HFBNickel Nickel

densitiesdensities




Black solid line : Black solid line : HFB boxHFB box

Dashed green line :Dashed green line : GHFB coord. GHFB coord.

Dotted blue line :Dotted blue line : GHF basisGHF basis

Gamow HFBGamow HFBNickel Nickel pairingpairing

densitiesdensities




Conclusion and perspectivesConclusion and perspectives• GSMGSM : Standard shell model : Standard shell model simplicitysimplicity and and powerpower beyond Hilbert space. beyond Hilbert space. CompleteComplete description of description of looselyloosely boundbound and and resonantresonant states.. states.. Spectroscopic factors:Spectroscopic factors: continum coupling necessary continum coupling necessary Cusps:Cusps: clear l-dependence clear l-dependence Important differences Important differences between between protonproton and and neutronneutron case case

• HFB expansions with Gamow and PTG bases:HFB expansions with Gamow and PTG bases: Precise tool Precise tool to study dripline to study dripline medium and heavy nucleimedium and heavy nuclei ContinuumContinuum fully taken into accountfully taken into account PTG basis PTG basis very good for very good for weakly bound systemsweakly bound systems Nickel chain Nickel chain close to neutron driplineclose to neutron dripline Deformed nuclei Deformed nuclei with Mg and Zn, with Mg and Zn, prolate and oblate deformation prolate and oblate deformation • PerspectivesPerspectives GSM:GSM: Coherent framework Coherent framework unifying unifying structurestructure and and reactionreaction,, many body many body open channels.open channels. Realistic interactions Realistic interactions and and DMRG method DMRG method to be applied in calculations.to be applied in calculations. PTG/HFB, Gamow/HFB: PTG/HFB, Gamow/HFB: mean field mean field and and beyond (QRPA)beyond (QRPA)

Description of drip-line nuclei with GSM and Gamow/HFB frameworks

Documents

Transcript of Description of drip-line nuclei with GSM and Gamow/HFB frameworks